Attempting to stretch texture towards edges
Set up the nodes as pictured below. In Texture Coordinate node choose the Normal output. Select the Normal option in a Mapping node and experiment with its Scale settings.


Change the Z Scale to increase the stretch of a noise.


I used this technique to procedurally create the iris. Hope this is the effect you're looking for.




UPDATE: After playing with the nodes I found out a new solution, though I am not sure it's the proper way of doing it.